Learning Riemannian Metrics from Gravitational Waves

Ref Bari, Brown University

Advisor: Prof. Brendan Keith

Acknowledgements

Brendan Keith

Scott Field

Collin Capano

Michael Pürrer

Pranav Vinod

Morgan Beck

Acknowledgements

Neural ODE DynAMO

NSF Award 2407452

For Data-Driven Discovery of

Astrophysical Models and Orbits

Motivation

Motivation

Motivation

Event Horizon

Singularity

Motivation

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

Motivation

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

Motivation

for decisive contributions to the LIGO detector and the observation of gravitational waves

2017 Nobel Prize in Physics

Question

h(t)

r(t),\phi(t)

g_{\mu\nu}

Answer

h(t)

r(t),\phi(t)

g_{\mu\nu}

Summary

Conservative Dynamics: We created a NN which can learn solutions to Einstein's Field Equations (Schwarzschild Metric) from only gravitational waves

Dissipative Dynamics: NN can learn both conservative and dissipative dynamics of a binary black hole inspiral for the synthetic dissipation case.

Approach

The Forward Problem

The Inverse Problem

Approach

The Inverse Problem

The Forward Problem

Metric

Orbits

Waveforms

Approach

The Forward Problem

The Inverse Problem

Metric

Orbits

Waveforms

GENERIC Formalism

Einstein's Field Equations

GENERIC Formalism

Synthetic Dissipation

Simulated Dissipation

Birkhoff's Theorem

Ricci Tensor

Approach

The Forward Problem

The Inverse Problem

Approach

The Forward Problem

The Inverse Problem

\{x(t),y(t)\} \to h(t)

Approach

The Forward Problem

\{x(t),y(t)\} \to h(t)

The Inverse Problem

h(t)\to \{x(t),y(t)\}

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

m_1

m_2

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

m_1

m_2

m_1

m_2

M=m_1+m_2

\vec{r}_1=\frac{m_2}{M}\vec{r}

\vec{r}_2=-\frac{m_1}{M}\vec{r}

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

The Forward Problem

Metric

Orbits

Waveforms

Credit: Wikipedia, Schwarzschild Geodesics

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

The Forward Problem

Metric

Orbits

Waveforms

g_{\mu\nu}

\vec{x}_i

\{\dot r, \dot \phi, \dot p_r, \dot p_\phi \}

h(t)

State Variables

\phi

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

The Forward Problem

Metric

Orbits

Waveforms

g_{\mu\nu}

\vec{x}_i

\{\dot r, \dot \phi, \dot p_r, \dot p_\phi \}

h(t)

Metric

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

The Forward Problem

Metric

Orbits

Waveforms

g_{\mu\nu}

\vec{x}_i

\{\dot r, \dot \phi, \dot p_r, \dot p_\phi \}

h(t)

Hamiltonian

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

The Forward Problem

Metric

Orbits

Waveforms

g_{\mu\nu}

\vec{x}_i

\{\dot r, \dot \phi, \dot p_r, \dot p_\phi \}

h(t)

Equations of Motion

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

The Forward Problem

Metric

Orbits

Waveforms

g_{\mu\nu}

\vec{x}_i

\{\dot r, \dot \phi, \dot p_r, \dot p_\phi \}

h(t)

Gravitational Waveform

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

The Forward Problem

Metric

Orbits

Waveforms

g_{\mu\nu}= \begin{pmatrix} -(1-\frac{2}{r}) & 0 & 0 & 0 \\ 0 & (1-\frac{2}{r})^{-1} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2\theta \end{pmatrix}

Schwarzschild Metric

\phi

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

The Forward Problem

Metric

Orbits

Waveforms

g_{\mu\nu}= \begin{pmatrix} -(1-\frac{2}{r}) & 0 & 0 & 0 \\ 0 & (1-\frac{2}{r})^{-1} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \end{pmatrix}

Schwarzschild Metric

\phi

p^{\mu}

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

The Forward Problem

Metric

Orbits

Waveforms

\phi

p_{\mu}=(p^t, p^r, p^\theta, p^\phi)

g_{\mu\nu}= \begin{pmatrix} -(1-\frac{2}{r}) & 0 & 0 & 0 \\ 0 & (1-\frac{2}{r})^{-1} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \end{pmatrix}

Schwarzschild Metric

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

The Forward Problem

Metric

Orbits

Waveforms

\phi

H = \frac{1}{2}p_{\mu} g^{-1}_{\mu\nu} p_{\nu}

g^{-1}_{\mu\nu}

Hamiltonian

p_{\nu}

p_{\mu}

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

The Forward Problem

Metric

Orbits

Waveforms

\phi

H = \frac{1}{2}\left[-\left(1-\frac{2M}{r}\right)^{-1}(p_t)^2+ \left(1-\frac{2M}{r}\right)(p_r)^2+ r^{-2}(p_\phi)^2\right]

Hamiltonian

H = \frac{1}{2}\left[-\left(1-\frac{2M}{r}\right)^{-1}(p_t)^2+ \left(1-\frac{2M}{r}\right)(p_r)^2+ r^{-2}(p_\phi)^2\right]

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

\phi

\dot{q}=\frac{\partial H}{\partial p}

Hamiltonian Equations

\dot{p}=-\frac{\partial H}{\partial q}

The Forward Problem

Metric

Orbits

Waveforms

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

\phi

\dot{r}=g^{rr}p_r

\dot{p}_r=-\frac{\partial H}{\partial r}

The Forward Problem

Metric

Orbits

Waveforms

\dot{\phi} = g^{\phi\phi}p_\phi

\dot{p_\phi} =\dot{L}

r Equations of Motion

Φ Equations of Motion

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

\phi

The Forward Problem

Metric

Orbits

Waveforms

h(t)=\frac{\ddot{Q}}{R}

= \int \rho (r_ir_j-\frac{1}{3}r^2 \delta_{ij})

Q_{ij}

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

\phi

The Forward Problem

Metric

Orbits

Waveforms

h(t)=\frac{\ddot{Q}}{R}

\sim mr^2\sin(\omega t)

Q_{ij}

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

\phi

The Forward Problem

Metric

Orbits

Waveforms

h(t)=\frac{\ddot{Q}}{R}

\sim mr^2 \omega^2 \sin(\omega t)

\ddot{Q}_{ij}

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

The Forward Problem

Metric

Orbits

Waveforms

\sim mr^2 \omega^2 \sin(\omega t)

\ddot{Q}_{ij}

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

The Forward Problem

Metric

Orbits

Waveforms

\sim mr^2 \omega^2 \sin(\omega t)

\ddot{Q}_{ij}

\dot{p_\phi} =\dot{L}-\Phi_L

Angular Momentum Flux

Forward Problem

The Inverse Problem

The Imaging Problem

h(t)\to \{x(t),y(t)\}

T_{res}=P/4

Metric

Orbits

Waveforms

Conservative Dynamics

The Forward Problem

Dissipative Dynamics

Approach

The Forward Problem

\{x(t),y(t)\} \to h(t)

The Inverse Problem

h(t)\to \{x(t),y(t)\}

Approach

The Inverse Problem

GENERIC Formalism

Einstein's Field Equations

GENERIC Formalism

Synthetic Dissipation

Birkhoff's Theorem

Ricci Tensor

h(t)

r(t),\phi(t)

g_{\mu\nu}

Approach

The Inverse Problem

GENERIC Formalism

Einstein's Field Equations

GENERIC Formalism

Synthetic Dissipation

Simulated Dissipation

Birkhoff's Theorem

Ricci Tensor

dU = \underbrace{dW} +\underbrace{dQ}

conservative

dissipative

Equilibrium Thermodynamics

Approach

The Inverse Problem

GENERIC Formalism

Einstein's Field Equations

GENERIC Formalism

Synthetic Dissipation

Simulated Dissipation

Birkhoff's Theorem

Ricci Tensor

dU = \underbrace{dW} +\underbrace{dQ}

conservative

dissipative

dU = {-p dV} +{T dS}

Equilibrium Thermodynamics

Approach

The Inverse Problem

GENERIC Formalism

Einstein's Field Equations

GENERIC Formalism

Synthetic Dissipation

Simulated Dissipation

Birkhoff's Theorem

Ricci Tensor

Inequilibrium Thermodynamics

\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}

\dot{x} = L \nabla E + M \nabla S

conservative

dissipative

\nabla E

Approach

The Inverse Problem

GENERIC Formalism

Einstein's Field Equations

GENERIC Formalism

Synthetic Dissipation

Simulated Dissipation

Birkhoff's Theorem

Ricci Tensor

Inequilibrium Thermodynamics

\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}

\dot{x} = L \nabla E + M \nabla S

conservative

dissipative

\begin{pmatrix} \dot{r} \\ \dot{\phi} \\ \dot{p_r} \\ \dot{p_\phi} \end{pmatrix} = \begin{pmatrix} 0 & I\\ -I & 0 \end{pmatrix} \begin{pmatrix} \partial H / \partial {r} \\ \partial H / \partial {\phi} \\ \partial H / \partial {p_r} \\ \partial H / \partial {p_\phi} \end{pmatrix}

\nabla E

\dot{x}

Approach

The Inverse Problem

GENERIC Formalism

Einstein's Field Equations

GENERIC Formalism

Synthetic Dissipation

Simulated Dissipation

Birkhoff's Theorem

Ricci Tensor

Inequilibrium Thermodynamics

\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}

\dot{x} = L \nabla E + M \nabla S

conservative

dissipative

\begin{pmatrix} \dot{r} \\ \dot{\phi} \\ \dot{p_r} \\ \dot{p_\phi} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ -\Phi_L \end{pmatrix}

M\nabla S

\dot{x}

Approach

The Inverse Problem

GENERIC Formalism

Einstein's Field Equations

GENERIC Formalism

Synthetic Dissipation

Simulated Dissipation

Birkhoff's Theorem

Ricci Tensor

Inequilibrium Thermodynamics

\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}

\dot{x} = L \nabla E + M \nabla S

conservative

dissipative

\nabla E

\dot{x}

M\nabla S

Approach

The Inverse Problem

GENERIC Formalism

Einstein's Field Equations

GENERIC Formalism

Synthetic Dissipation

Simulated Dissipation

Birkhoff's Theorem

Ricci Tensor

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}

Approach

The Inverse Problem

GENERIC Formalism

Einstein's Field Equations

GENERIC Formalism

Synthetic Dissipation

Simulated Dissipation

Birkhoff's Theorem

Ricci Tensor

f(g_{\mu\nu})=T_{\mu\nu}

g_{\text{Kerr}}

g_{\text{Schwarzschild}}

g_{\text{Newtonian}}

g_{\text{Minkowski}}

Approach

The Inverse Problem

GENERIC Formalism

Einstein's Field Equations

GENERIC Formalism

Synthetic Dissipation

Simulated Dissipation

Birkhoff's Theorem

Ricci Tensor

g_{\text{Kerr}}

g_{\text{Schwarzschild}}

g_{\text{Newtonian}}

g_{\text{Minkowski}}

Approach

The Inverse Problem

GENERIC Formalism

Einstein's Field Equations

GENERIC Formalism

Synthetic Dissipation

Simulated Dissipation

Birkhoff's Theorem

Ricci Tensor

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}

If a metric is a solution to

Einstein's Field Equations,

then R = 0

R^{\mu\nu}R_{\mu\nu}=0

Approach

The Inverse Problem

GENERIC Formalism

Einstein's Field Equations

GENERIC Formalism

Synthetic Dissipation

Simulated Dissipation

Birkhoff's Theorem

Ricci Tensor

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}

R-\frac{1}{2}R\cdot 4 = 0

g^{\mu\nu}(R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=0)

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=0

-2R = 0 \to R = 0

The Inverse Problem

GENERIC Formalism

Einstein's Field Equations

GENERIC Formalism

Synthetic Dissipation

Simulated Dissipation

Birkhoff's Theorem

Ricci Tensor

Approach

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}

The unique spherically symmetric solution to Einstein's Field Equations is a static and asymptotically flat metric"

On the discovery of Birkhoff's theorem (Johansen, F Ravndal, 2005)

g_{\mu\nu}

\Gamma^{\alpha}_{ \mu\nu}

R^{\alpha}_{ \mu\nu\rho}

R_{ \mu\nu}

Metric

Christoffel Symbols

Riemann Curvature Tensor

Ricci Curvature Tensor

Ricci Scalar

Einstein's Field Equations: A Brief Intro

g_{\mu\nu}

\Gamma^{\alpha}_{ \mu\nu}

R^{\alpha}_{ \mu\nu\rho}

R_{ \mu\nu}

Metric

Christoffel Symbols

Riemann Curvature Tensor

Ricci Curvature Tensor

Ricci Scalar

Einstein's Field Equations: A Brief Intro

Coordinate-Dependent!

g_{\mu\nu}

\Gamma^{\alpha}_{ \mu\nu}

R^{\alpha}_{ \mu\nu\rho}

R_{ \mu\nu}

Metric

Christoffel Symbols

Riemann Curvature Tensor

Ricci Curvature Tensor

Ricci Scalar

Einstein's Field Equations: A Brief Intro

Coordinate-Independent!

g_{\mu\nu}

\Gamma^{\alpha}_{ \mu\nu}

R^{\alpha}_{ \mu\nu\rho}

R_{ \mu\nu}

R_{\beta \mu \nu}^\alpha=\Gamma_{\beta \nu, \mu}^\alpha-\Gamma_{\beta \mu, \nu}^\alpha+\Gamma_{\mu \rho}^\alpha \Gamma_{\beta \nu}^\rho-\Gamma_{\nu \rho}^\alpha \Gamma_{\beta \mu}^\rho

R_{\mu\nu}=R^{\alpha}_{\mu\alpha\nu}

R = g^{\mu\nu}R_{\mu\nu}

Metric

Christoffel Symbols

Riemann Curvature Tensor

Ricci Curvature Tensor

Ricci Scalar

\Gamma_{\mu \nu}^\alpha=\frac{1}{2} g^{\alpha \beta}\left(g_{\beta \mu, \nu}+g_{\beta \nu, \mu}-g_{\mu \nu, \beta}\right)

Einstein's Field Equations: A Brief Intro

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}

g_{\mu\nu}

\Gamma^{\alpha}_{ \mu\nu}

R^{\alpha}_{ \mu\nu\rho}

R_{ \mu\nu}

R_{\beta \mu \nu}^\alpha=\Gamma_{\beta \nu, \mu}^\alpha-\Gamma_{\beta \mu, \nu}^\alpha+\Gamma_{\mu \rho}^\alpha \Gamma_{\beta \nu}^\rho-\Gamma_{\nu \rho}^\alpha \Gamma_{\beta \mu}^\rho

R_{\mu\nu}=R^{\alpha}_{\mu\alpha\nu}

R = g^{\mu\nu}R_{\mu\nu}

Metric

Christoffel Symbols

Riemann Curvature Tensor

Ricci Curvature Tensor

Ricci Scalar

\Gamma_{\mu \nu}^\alpha=\frac{1}{2} g^{\alpha \beta}\left(g_{\beta \mu, \nu}+g_{\beta \nu, \mu}-g_{\mu \nu, \beta}\right)

Einstein's Field Equations: A Brief Intro

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}

g_{\mu\nu}

\Gamma^{\alpha}_{ \mu\nu}

R^{\alpha}_{ \mu\nu\rho}

R_{ \mu\nu}

R_{\beta \mu \nu}^\alpha=\Gamma_{\beta \nu, \mu}^\alpha-\Gamma_{\beta \mu, \nu}^\alpha+\Gamma_{\mu \rho}^\alpha \Gamma_{\beta \nu}^\rho-\Gamma_{\nu \rho}^\alpha \Gamma_{\beta \mu}^\rho

R_{\mu\nu}=R^{\alpha}_{\mu\alpha\nu}

R = g^{\mu\nu}R_{\mu\nu}

Metric

Christoffel Symbols

Riemann Curvature Tensor

Ricci Curvature Tensor

Ricci Scalar

Einstein's Field Equations: A Brief Intro

\Gamma_{\mu \nu}^\alpha=\frac{1}{2} g^{\alpha \beta}\left(g_{\beta \mu, \nu}+g_{\beta \nu, \mu}-g_{\mu \nu, \beta}\right)

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}

g_{\mu\nu}

\Gamma^{\alpha}_{ \mu\nu}

R^{\alpha}_{ \mu\nu\rho}

R_{ \mu\nu}

R_{\beta \mu \nu}^\alpha=\Gamma_{\beta \nu, \mu}^\alpha-\Gamma_{\beta \mu, \nu}^\alpha+\Gamma_{\mu \rho}^\alpha \Gamma_{\beta \nu}^\rho-\Gamma_{\nu \rho}^\alpha \Gamma_{\beta \mu}^\rho

R_{\mu\nu}=R^{\alpha}_{\mu\alpha\nu}

R = g^{\mu\nu}R_{\mu\nu}

Metric

Christoffel Symbols

Riemann Curvature Tensor

Ricci Curvature Tensor

Ricci Scalar

Einstein's Field Equations: A Brief Intro

\Gamma_{\mu \nu}^\alpha=\frac{1}{2} g^{\alpha \beta}\left(g_{\beta \mu, \nu}+g_{\beta \nu, \mu}-g_{\mu \nu, \beta}\right)

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}

g_{\mu\nu}

\Gamma^{\alpha}_{ \mu\nu}

R^{\alpha}_{ \mu\nu\rho}

R_{ \mu\nu}

R_{\beta \mu \nu}^\alpha=\Gamma_{\beta \nu, \mu}^\alpha-\Gamma_{\beta \mu, \nu}^\alpha+\Gamma_{\mu \rho}^\alpha \Gamma_{\beta \nu}^\rho-\Gamma_{\nu \rho}^\alpha \Gamma_{\beta \mu}^\rho

R_{\mu\nu}=R^{\alpha}_{\mu\alpha\nu}

R = g^{\mu\nu}R_{\mu\nu}

Metric

Christoffel Symbols

Riemann Curvature Tensor

Ricci Curvature Tensor

Ricci Scalar

Einstein's Field Equations: A Brief Intro

0 for vacuum

solutions of

field equations!

\Gamma_{\mu \nu}^\alpha=\frac{1}{2} g^{\alpha \beta}\left(g_{\beta \mu, \nu}+g_{\beta \nu, \mu}-g_{\mu \nu, \beta}\right)

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi G

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi G T_{\mu\nu}

T_{\mu\nu}(x^{\mu})\neq0

g_{\mu\nu}(x^{\mu})\neq0

R_{\mu\nu}(x^{\mu})\neq0

x^{\mu}=(t_0, r_0, \theta_0, \phi_0)

x^{\mu}=(t_1, r_1, \theta_1, \phi_1)

T_{\mu\nu}(x^{\mu})=0

g_{\mu\nu}(x^{\mu})\neq0

R_{\mu\nu}(x^{\mu})=0

Einstein's Field Equations: A Brief Intro

Approach

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}

The Inverse Problem

GENERIC Formalism

Einstein's Field Equations

GENERIC Formalism

Synthetic Dissipation

Simulated Dissipation

Birkhoff's Theorem

Ricci Tensor

R^{\mu\nu}R_{ \mu\nu}=0

if Neural Network

predicts correct solution

to Einstein's Field Equations!

h(t)\to \{x(t),y(t)\}

H_{Minkowski} = -p_t^2 + p_r^2 + r^2 p_\theta^2+r^2\sin^2\theta p_\phi^2

H_{Schwarzschild} = - \left(1-\frac{2M}{r}\right)^{-1}\frac{p_t^2}{2} + \left(1-\frac{2M}{r}\right) \frac{p_r^2}{2} + \frac{p_φ^2}{2r^2}

Neural ODE

Approach

h(t)\to \{x(t),y(t)\}

H_{Schwarzschild} = - \left(1-\frac{2M}{r}\right)^{-1}\frac{p_t^2}{2} + \left(1-\frac{2M}{r}\right) \frac{p_r^2}{2} + \frac{p_φ^2}{2r^2}

Neural ODE:

Training Data

\dot x=L\nabla E \to \dot x = J \nabla H_{total}

Approach

H_{Minkowski} = -p_t^2 + p_r^2 + r^2 p_\theta^2+r^2\sin^2\theta p_\phi^2

h(t)\to \{x(t),y(t)\}

H_{Minkowski} = -p_t^2 + p_r^2 + r^2 p_\theta^2+r^2\sin^2\theta p_\phi^2

H_{Schwarzschild} = - \left(1-\frac{2M}{r}\right)^{-1}\frac{p_t^2}{2} + \left(1-\frac{2M}{r}\right) \frac{p_r^2}{2} + \frac{p_φ^2}{2r^2}

Neural ODE:

Training Data

\dot x=L\nabla E \to \dot x = J \nabla H_{total} \to \dot x = J \nabla (H_{Mink} + f_{NN}(\theta))

Approach

h(t)\to \{x(t),y(t)\}

H_{Minkowski} = -p_t^2 + p_r^2 + r^2 p_\theta^2+r^2\sin^2\theta p_\phi^2

H_{Schwarzschild} = - \left(1-\frac{2M}{r}\right)^{-1}\frac{p_t^2}{2} + \left(1-\frac{2M}{r}\right) \frac{p_r^2}{2} + \frac{p_φ^2}{2r^2}

Neural ODE:

Training Data

\dot x=L\nabla E \to \dot x = J \nabla H_{total} \to \dot x = J \nabla (H_{Mink} + f_{NN}(\theta))

ODE Solver:

u(t) = [t(t), r(t), \theta(t), \phi(t), p_t(t), p_r(t), p_\theta(t), p_\phi(t)]

Approach

h(t)\to \{x(t),y(t)\}

H_{Minkowski} = -p_t^2 + p_r^2 + r^2 p_\theta^2+r^2\sin^2\theta p_\phi^2

H_{Schwarzschild} = - \left(1-\frac{2M}{r}\right)^{-1}\frac{p_t^2}{2} + \left(1-\frac{2M}{r}\right) \frac{p_r^2}{2} + \frac{p_φ^2}{2r^2}

Neural ODE:

Training Data

\dot x=L\nabla E \to \dot x = J \nabla H_{total} \to \dot x = J \nabla (H_{Mink} + f_{NN}(\theta))

ODE Solver:

u(t) = [t(t), r(t), \theta(t), \phi(t), p_t(t), p_r(t), p_\theta(t), p_\phi(t)]

(r(t), \phi(t)) \to (x(t), y(t)) \to h(t)

Approach

H_{Minkowski} = -p_t^2 + p_r^2 + r^2 p_\theta^2+r^2\sin^2\theta p_\phi^2

H_{Schwarzschild} = - \left(1-\frac{2M}{r}\right)^{-1}\frac{p_t^2}{2} + \left(1-\frac{2M}{r}\right) \frac{p_r^2}{2} + \frac{p_φ^2}{2r^2}

Neural ODE:

Training Data

\dot x=L\nabla E \to \dot x = J \nabla H_{total} \to \dot x = J \nabla (H_{Mink} + f_{NN}(\theta))

ODE Solver:

u(t) = [t(t), r(t), \theta(t), \phi(t), p_t(t), p_r(t), p_\theta(t), p_\phi(t)]

(r(t), \phi(t)) \to (x(t), y(t)) \to h(t)

BFGS Optimizer:

\begin{align*}\text{min}_{\eta} \sum &(h_{pred}-h_{true})^2 + \text{(NN\_params)}^2+R^{\mu\nu}R_{\mu\nu} \end{align*}

Neural ODE:

Training Data:

H_{pred} = \frac{1}{2}p^T g_{pred} p = \frac{1}{2}p^T [g_{Mink}+f_{NN}] p \to \dot x = L\nabla H_{pred}

ODE Solver:

u(t) = [t(t), r(t), \theta(t), \phi(t), p_t(t), p_r(t), p_\theta(t), p_\phi(t)]

(r(t), \phi(t)) \to (x(t), y(t)) \to h(t)

BFGS Optimizer:

\begin{align*}\text{min}_{\eta} \sum &(h_{pred}-h_{true})^2 + \text{(NN\_params)}^2+R^{\mu\nu}R_{\mu\nu} \end{align*}

\frac{du}{d\tau} = L \nabla H_{Schwarzschild} \to \frac{du}{dt} = \frac{du}{d\tau}\frac{d\tau}{dt} \to u(t) \to h(t)

g_{Minkowski}=\begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^{2}\sin^2\theta \end{pmatrix}

g_{Schwarzschild} = \begin{pmatrix} -\left(1-\frac{2M}{r} \right) & 0 & 0 & 0 \\ 0 & \left(1-\frac{2M}{r} \right)^{-1} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \end{pmatrix}